Parallel And Perpendicular Axes Theorems And Their Applications

Moment of Inertia: In rotational motion, the moment of inertia is the magnitude by which a rotating object opposes its motion. Therefore, it is also referred to as the rotational inertia of the object. 

If we speak from the point of view of maths, it can be expressed as the product of the distance of the object from the axis and its mass. The rotational inertia is usually written as L in mathematical formulas and relations. 

The unit of rotational moment of inertia is kg/m2. The moment of inertia of a rigid object rotating about its axis can be given by the below formula:

I = m1r12+m2r22+m3r32+…………… = i =1nmiri2

Here, m1, m2, and m3 are the masses of any three particles of the rigid object, and r1, r2, and r3 are their respective distances from the axis of rotation.  

When we speak about rotational motion in physics, two theorems are of utmost importance. These theorems are perpendicular and parallel axis theorems. Let’s discuss them one by one. 

Parallel Axis Theorem 

The Parallel Axis Theorem states that an object’s moment of inertia at any axis is the same as the addition of the moment of inertia at a parallel axis with perpendicular distance’s square (the distance which is between the two axes) and mass of the object. However, for this to be true, the parallel axis should go via its centre of mass. The below formula will elaborate this definition better:

I = ICM +Mr2

Here, a moment of inertia at the parallel axis is denoted by ICM, and r is the perpendicular length between the two parallel axes. Also, M is the object’s total mass.  

Perpendicular Axis Theorem 

Suppose that a two-dimensional object has an axis Iz that is perpendicular to its plane. Now, the moment of inertia at Iz is the same as the addition of the object’s moments of inertia at the two perpendicular axes that lie on the same plane, intersect each other (at the same point at which Iz goes through the object), and are mutually perpendicular to one another. To elaborate further, look at the below equation:

Iz = Ix + Iy

Here, X and Y axes intersect each other at the point through which the perpendicular axis Iz passes through the object. Ix and Iy are the moments of inertia at the X and Y-axis, respectively. 

The perpendicular axis theorem is mostly applicable to 1-dimensional objects or laminar types of objects. On the other hand, the parallel axis theorem applies to 2 or 3-dimensional rigid objects. 

The below table contains the formula for different homogeneous rigid objects and their moment of inertia. These formulas will help the students to solve problems based on the moment of inertia of objects of different shapes and sizes:

Conclusion 

The moment of inertia has a vital role in determining the torque required to attain the specific angular acceleration in the rotational axis. In this article, we discussed the moment of inertia formula and the impact of moi on the system of particles and rigid bodies, the parallel axis theorem, and perpendicular axis theorem, and radius of gyration. The moment of inertia is a critical topic in physics that plays an essential role in several entrance examinations.